If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.

My question is regarding accessing data in non-trivial cohomological degree of a cosimplicial space. Ultimately, for me, this comes down to questions about looping and delooping cosimplicial spaces and/or quasicategories.

1. Given a cosimplicial quasicategory $C^\bullet$, can I take the homotopy pullback of $\ast\to C^\bullet\leftarrow \ast$ (with some chosen base point in $C^\bullet$), to obtain a notion of $\Omega C^\bullet$?

2. If so, can I then compute what should correspond to data in cohomological degree 1, and hence in $\pi_{-1}Tot(\Omega C^\bullet)$, of the connected component containing the base point, by somehow "delooping" $\Omega C^\bullet$ and attempting to compute $\pi_0Tot(B\Omega C^\bullet)$?

My main issue is what the construction of "delooping" should be in the situation of a 1-fold looped cosimplicial quasicategory? Also, does the category of cosimplicial quasicategories admit an initial object?

The motivation for this question comes from homotopical descent theory.

Thanks, as usual!

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.

My question is regarding accessing data in non-trivial cohomological degree of a cosimplicial space. Ultimately, for me, this comes down to questions about looping and delooping cosimplicial spaces and/or quasicategories.

1. Given a cosimplicial quasicategory $C^\bullet$, can I take the homotopy pullback of $\ast\to C^\bullet\leftarrow \ast$ (with some chosen base point in $C^\bullet$), to obtain a notion of $\Omega C^\bullet$?

2. If so, can I then compute what should correspond to data in cohomological degree 1, and hence what would be in $\pi_{-1}Tot(\Omega C^\bullet)$ (using e.g. a Bousfield Kan sort of spectral sequence $\pi^s\pi_t\Rightarrow\pi_{t-s})$, of the connected component containing the base point, by somehow "delooping" $\Omega C^\bullet$ and attempting to compute $\pi_0Tot(B\Omega C^\bullet)$? Of course, without stability, it doesn't really make sense to talk about $\pi_{-1}$, that's just where this "information would live" if it did.

My main issue is what the construction of "delooping" should be in the situation of a 1-fold looped cosimplicial quasicategory? Also, does the category of cosimplicial quasicategories admit an initial object?

The motivation for this question comes from homotopical descent theory.

Thanks, as usual!